Online Training

Unlike offline training which processes a complete batch of captured data, online training updates the DPD coefficient vector with each new sample or small block of samples. This recursive update mechanism eliminates the need to store large matrices and recompute the full solution from scratch. Algorithms like Recursive Least Squares (RLS) and Least Mean Squares (LMS) are specifically designed for this incremental processing paradigm, making them ideal for embedded DSP implementations where memory and latency are constrained.

Power amplifiers exhibit time-varying characteristics due to thermal memory effects, aging, bias drift, and changing operating conditions. Online training continuously adapts the predistorter to these variations by incorporating a forgetting factor that exponentially weights recent data more heavily than past observations. This enables the system to track slow changes in amplifier gain compression and phase distortion without requiring periodic recalibration cycles or interrupting transmission.

Online training operates within a closed-loop feedback architecture where the error between the desired linear output and the actual power amplifier output drives coefficient updates. In the Direct Learning Architecture (DLA), this error is computed from the PA output and fed back to the predistorter adaptation block. The algorithm iteratively minimizes this error signal, converging toward the optimal inverse nonlinearity that linearizes the amplifier chain.

Online algorithms present a spectrum of complexity-vs-convergence tradeoffs:

• LMS: O(N) complexity per iteration with slow convergence
• NLMS: O(N) complexity with improved stability under input power variations
• RLS: O(N²) complexity with order-of-magnitude faster convergence
• QR-RLS: O(N²) complexity with superior numerical stability using Givens rotations The choice depends on available DSP resources, required tracking speed, and the condition number of the input correlation matrix.

Continuous operation over extended periods exposes online algorithms to numerical instability risks. The recursive update of the inverse correlation matrix in RLS can accumulate round-off errors, potentially causing divergence. QR-RLS mitigates this by propagating the square-root of the inverse correlation matrix using orthogonal transformations, maintaining positive-definiteness. Regularization parameters are often injected periodically to prevent ill-conditioning when the input signal lacks persistent excitation.

The steady-state performance of online training is characterized by misadjustment — the excess mean squared error above the theoretical Wiener optimum. This arises from gradient noise in stochastic algorithms like LMS. A larger step size accelerates convergence rate but increases misadjustment, embodying the classic bias-variance tradeoff. Practical systems often employ variable step-size strategies that start large for rapid acquisition and decay for low steady-state error.

An adaptive algorithm that recursively updates the inverse of the input correlation matrix to achieve faster convergence than gradient-based methods. RLS minimizes a weighted least-squares cost function, making it ideal for tracking time-varying power amplifier characteristics in online DPD systems.

• Computational complexity: O(N²) per iteration
• Converges in approximately 2N iterations
• Requires a forgetting factor to track non-stationary signals

A foundational stochastic gradient descent algorithm that updates filter coefficients iteratively based on the instantaneous estimate of the mean squared error gradient. LMS offers low computational complexity at the cost of slower convergence compared to RLS.

• Computational complexity: O(N) per iteration
• Sensitive to eigenvalue spread of the input signal
• Step size μ controls the convergence-misadjustment tradeoff

A variant of the LMS algorithm that normalizes the step size by the power of the input signal vector. This normalization improves convergence stability in the presence of fluctuating signal levels, making it more robust for online DPD with varying envelope signals.

• Step size normalized by ||x(n)||² + ε
• More robust to input power variations than standard LMS
• Small regularization constant ε prevents division by zero

A numerically robust implementation of the Recursive Least Squares algorithm that uses Givens rotations to directly update the square-root of the inverse correlation matrix. QR-RLS avoids the numerical instability of the standard RLS by maintaining the Cholesky factor rather than the full correlation matrix.

• Uses orthogonal transformations for superior numerical stability
• Well-suited for fixed-point FPGA implementations
• Avoids covariance matrix ill-conditioning